By Equation (REF ), we know that
the Green function \(G\)  may be decomposed into
a `singular part', which blows up logarithmically
near the diagonal of \(M\times M\) , and a `regular part' \(H\in C^1(M\times M)\) .
Since \(G\)  is symmetric in \(x\)  and \(y\) 
(see e.g. [1]}), we deduce
\(H(x, \, y) = H(y, \, x)\) 
